Central Tendency

Central tendency refers to measures that identify the center of a dataset.
Continuous and ordinal variables:

• Mean: average value.
  
• Median: value at ≥50% of the distribution.
  
• Mode: more frequently occurring value.¹

Categorical variables: report as tables or recode in binary.

¹ This value is often meaningless for continuous variables, so is rarely included.

Mean

\[ \bar{Y} = \frac{\sum_{i=1}^n Y_i}{n} \]

Components:

• ({Y}) (“Y-bar”) = mean of (Y)
  
• (_{i=1}^n Y_i) = the sum of all values of (Y)
  
• (n) = the total number of observations

Properties of Mean

1. Zero-Sum Property

\[ \sum_{i=1}^n (Y_i - \bar{Y}) = 0 \]

2. Least-Squares Property
\[ \sum_{i=1}^n (Y_i - \bar{Y})^2 \;<\; \sum_{i=1}^n (Y_i - c)^2 \quad \forall\; c \neq \bar{Y} \]

Median

The median is the middle value of a variable when the observations are ordered from smallest to largest.

It divides the distribution into two equal halves: 50% of values are below the median and 50% are above it.

• If there is an odd number of observations, the median is the middle value.
  
• If there is an even number of observations, the median is the average of the two middle values.

Key advantage: The median is resistant to outliers and skewed data.

Age

Age: Mean is 52.48

Age: Median is 56

Binary

Binary: Mean is 0.623

Binary: Median is 1

Mean vs. Median

Imagine there are 5 people sitting in a bar:

• Net worth: $10,000, $120,000, $140,000, $250,000
  
• Mean: $130,000
  
• Median: $130,000

Imagine that Elon Musk walks into the bar:

• Net worth: $10,000, $120,000, $140,000, $250,000, $393,000,000,000
  
• Mean: $78,600,104,000
  
• Median: $140,000

Variance

Variance is the typical (squared) distance between an observation and the mean:

\[ \mathrm{var}(Y) = s^2 = \frac{\sum_{i=1}^n (Y_i - \bar{Y})^2}{n} \]

• Calculate each observation’s distance from the mean: \((Y_i - \bar{Y})\)
  
• Square the distances: \(((Y_i - \bar{Y})^2)\)
  • Now they’re all positive.
    
  • Bigger differences matter more.
    
• Take the average of the squared differences.

Standard Deviation

\[ \mathrm{sd}(Y) = s = \sqrt{\frac{\sum_{i=1}^n (Y_i - \bar{Y})^2}{n}} \]

Assuming a normal distribution:

• 68% of observations will be within one standard deviation.
  
• 95% of observations will be within two standard deviations.
  
• 99.5% of observations will be within three standard deviations.

Age

Age: One Standard Deviation

Age: Two Standard Deviations

Age: Three Standard Deviations

Covariance

\[ \mathrm{cov}(X, Y) = \frac{\sum_{i=1}^n (X_i - \bar{X})(Y_i - \bar{Y})}{n} \] - If \(X\) and \(Y\) tend to increase together, most terms are positive → positive covariance.
- If there’s no clear relationship, positives and negatives cancel → covariance ≈ 0.

cov(Age, Weeks of Abortion) = -18.4

Correlation

\[ \mathrm{cor}(X,Y) = \frac{\mathrm{cov}(X,Y)}{\mathrm{sd}(X)\,\mathrm{sd}(Y)} \]

• \((\mathrm{cor}(X,Y)=1)\) → perfect positive linear relationship.
  
• \((\mathrm{cor}(X,Y)=-1)\) → perfect negative linear relationship.
  
• \((\mathrm{cor}(X,Y)=0)\) → no linear relationship.

cor(Age, Weeks of Abortion) = -0.086

Correlation

Why Visualize Data?

• Helps identify patterns and outliers
  
• Makes data easier to interpret
  
• Communicates findings effectively

Histogram

Histograms?

Histograms allow us to visualize the distribution of a single continuous variable.

• The x-axis displays the bins or intervals of the variable.
  
• The y-axis shows how many observations fall into each bin.

Box Plots

Box Plots?

Box plots help us compare the distributions of a continuous variable across categories.

• The continuous variable is on the y-axis.
  
• The categorical variable is on the x-axis.

Scatter Plots

Scatter Plots?

We use scatter plots to visualize the relationship between two continuous variables:

• The independent variable is on the x-axis.
  
• The dependent variable is on the y-axis.